3.3.75 \(\int \frac {\sqrt {\frac {e (a+b x^2)}{c+d x^2}}}{x^6} \, dx\) [275]

3.3.75.1 Optimal result

Integrand size = 26, antiderivative size = 424 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=-\frac {d \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{15 a^2 c^3}-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c x^5}-\frac {(b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 a c^2 x^3}+\frac {\left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{15 a^2 c^3 x}+\frac {\sqrt {d} \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^2 c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {d} (b c-4 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 a^2 c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

output

-1/15*d*(-8*a^2*d^2+3*a*b*c*d+2*b^2*c^2)*x*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a 
^2/c^3-1/5*(d*x^2+c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c/x^5-1/15*(-4*a*d+b*c) 
*(d*x^2+c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a/c^2/x^3+1/15*(-8*a^2*d^2+3*a*b* 
c*d+2*b^2*c^2)*(d*x^2+c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a^2/c^3/x+1/15*(-8* 
a^2*d^2+3*a*b*c*d+2*b^2*c^2)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*Ellip 
ticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*d^(1/2)*(e*(b* 
x^2+a)/(d*x^2+c))^(1/2)/a^2/c^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)-1/15*b 
*(-4*a*d+b*c)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/ 
c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*d^(1/2)*(e*(b*x^2+a)/(d*x^2+c 
))^(1/2)/a^2/c^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)
 

3.3.75.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.79 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-2 b^2 c^2 x^4+a b c x^2 \left (c-3 d x^2\right )+a^2 \left (3 c^2-4 c d x^2+8 d^2 x^4\right )\right )+i b c \left (-2 b^2 c^2-3 a b c d+8 a^2 d^2\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-2 i b c \left (-b^2 c^2-a b c d+2 a^2 d^2\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{15 a^2 \sqrt {\frac {b}{a}} c^3 x^5 \left (a+b x^2\right )} \]

input

Integrate[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^6,x]
 

output

-1/15*(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2 
)*(-2*b^2*c^2*x^4 + a*b*c*x^2*(c - 3*d*x^2) + a^2*(3*c^2 - 4*c*d*x^2 + 8*d 
^2*x^4)) + I*b*c*(-2*b^2*c^2 - 3*a*b*c*d + 8*a^2*d^2)*x^5*Sqrt[1 + (b*x^2) 
/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - ( 
2*I)*b*c*(-(b^2*c^2) - a*b*c*d + 2*a^2*d^2)*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 
 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(a^2*Sqrt[b 
/a]*c^3*x^5*(a + b*x^2))
 

3.3.75.3 Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2058, 377, 445, 445, 25, 27, 406, 320, 388, 313}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx\)

\(\Big \downarrow \) 2058

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \int \frac {\sqrt {b x^2+a}}{x^6 \sqrt {d x^2+c}}dx}{\sqrt {a+b x^2}}\)

\(\Big \downarrow \) 377

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {\int \frac {-3 b d x^2+b c-4 a d}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\int \frac {2 b^2 c^2+3 a b d c-8 a^2 d^2+b d (b c-4 a d) x^2}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {-\frac {\int -\frac {b d \left (\left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) x^2+a c (b c-4 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\frac {\int \frac {b d \left (\left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) x^2+a c (b c-4 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\frac {b d \int \frac {\left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) x^2+a c (b c-4 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\frac {b d \left (\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (b c-4 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\frac {b d \left (\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\frac {b d \left (\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\frac {b d \left (\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\)

input

Int[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^6,x]
 

output

(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(-1/5*(Sqrt[a + b*x^2]* 
Sqrt[c + d*x^2])/(c*x^5) + (-1/3*((b*c - 4*a*d)*Sqrt[a + b*x^2]*Sqrt[c + d 
*x^2])/(a*c*x^3) - (-((((2*b^2*c)/a + 3*b*d - (8*a*d^2)/c)*Sqrt[a + b*x^2] 
*Sqrt[c + d*x^2])/x) + (b*d*((2*b^2*c^2 + 3*a*b*c*d - 8*a^2*d^2)*((x*Sqrt[ 
a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTa 
n[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/ 
(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2)*(b*c - 4*a*d)*Sqrt[a + b*x^2 
]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[( 
c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/(a*c))/(3*a*c))/(5*c))) 
/Sqrt[a + b*x^2]
 

3.3.75.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 377

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*e*( 
m + 1))), x] - Simp[1/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) 
+ 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b 
*c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m 
, 2, p, q, x]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

3.3.75.4 Maple [A] (verified)

Time = 6.59 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.27

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \left (8 a^{2} d^{2} x^{4}-3 b d a c \,x^{4}-2 b^{2} c^{2} x^{4}-4 a^{2} c d \,x^{2}+a b \,c^{2} x^{2}+3 a^{2} c^{2}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{15 c^{3} x^{5} a^{2}}+\frac {b d \left (\frac {4 a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {a b \,c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 \left (8 a^{2} d^{2}-3 a b c d -2 b^{2} c^{2}\right ) a c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (e d a +e b c +e \left (a d -b c \right )\right )}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{15 a^{2} c^{3} \left (b \,x^{2}+a \right )}\)	\(539\)
default	\(-\frac {\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (8 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{3} x^{8}-3 \sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{2} x^{8}-2 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} d \,x^{8}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b c \,d^{2} x^{5}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{2} d \,x^{5}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{3} x^{5}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b c \,d^{2} x^{5}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{2} d \,x^{5}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{3} x^{5}+8 \sqrt {-\frac {b}{a}}\, a^{3} d^{3} x^{6}+\sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{2} x^{6}-4 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d \,x^{6}-2 \sqrt {-\frac {b}{a}}\, b^{3} c^{3} x^{6}+4 \sqrt {-\frac {b}{a}}\, a^{3} c \,d^{2} x^{4}-3 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{2} d \,x^{4}-\sqrt {-\frac {b}{a}}\, a \,b^{2} c^{3} x^{4}-\sqrt {-\frac {b}{a}}\, a^{3} c^{2} d \,x^{2}+4 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{3} x^{2}+3 \sqrt {-\frac {b}{a}}\, a^{3} c^{3}\right )}{15 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, c^{3} x^{5} a^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\)	\(708\)

input

int((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^6,x,method=_RETURNVERBOSE)
 

output

-1/15*(d*x^2+c)*(8*a^2*d^2*x^4-3*a*b*c*d*x^4-2*b^2*c^2*x^4-4*a^2*c*d*x^2+a 
*b*c^2*x^2+3*a^2*c^2)/c^3/x^5/a^2*(e*(b*x^2+a)/(d*x^2+c))^(1/2)+1/15/a^2*b 
*d/c^3*(4*a^2*c*d/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+1/c*d*x^2)^(1/2)/(b*d* 
e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d*e 
+b*c*e)/c/b/e)^(1/2))-a*b*c^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+1/c*d*x^2) 
^(1/2)/(b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)*EllipticF(x*(-b/a)^(1/2 
),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2))-2*(8*a^2*d^2-3*a*b*c*d-2*b^2*c^2)*a*c*e/ 
(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+1/c*d*x^2)^(1/2)/(b*d*e*x^4+a*d*e*x^2+b* 
c*e*x^2+a*c*e)^(1/2)/(e*d*a+e*b*c+e*(a*d-b*c))*(EllipticF(x*(-b/a)^(1/2),( 
-1+(a*d*e+b*c*e)/c/b/e)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d*e+b*c*e)/ 
c/b/e)^(1/2))))*(e*(b*x^2+a)/(d*x^2+c))^(1/2)*((d*x^2+c)*e*(b*x^2+a))^(1/2 
)/(b*x^2+a)
 

3.3.75.5 Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=-\frac {{\left (2 \, b^{3} c^{2} d + 3 \, a b^{2} c d^{2} - 8 \, a^{2} b d^{3}\right )} \sqrt {\frac {a c e}{d^{2}}} x^{5} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b^{3} c^{2} d + {\left (a^{2} b + 3 \, a b^{2}\right )} c d^{2} - 4 \, {\left (a^{3} + 2 \, a^{2} b\right )} d^{3}\right )} \sqrt {\frac {a c e}{d^{2}}} x^{5} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (2 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 8 \, a^{3} d^{3}\right )} x^{6} - 3 \, a^{3} c^{3} + 2 \, {\left (a b^{2} c^{3} + a^{2} b c^{2} d - 2 \, a^{3} c d^{2}\right )} x^{4} - {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{15 \, a^{3} c^{3} x^{5}} \]

input

integrate((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^6,x, algorithm="fricas")
 

output

-1/15*((2*b^3*c^2*d + 3*a*b^2*c*d^2 - 8*a^2*b*d^3)*sqrt(a*c*e/d^2)*x^5*sqr 
t(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (2*b^3*c^2*d + (a^2* 
b + 3*a*b^2)*c*d^2 - 4*(a^3 + 2*a^2*b)*d^3)*sqrt(a*c*e/d^2)*x^5*sqrt(-b/a) 
*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - ((2*a*b^2*c^2*d + 3*a^2*b*c 
*d^2 - 8*a^3*d^3)*x^6 - 3*a^3*c^3 + 2*(a*b^2*c^3 + a^2*b*c^2*d - 2*a^3*c*d 
^2)*x^4 - (a^2*b*c^3 - a^3*c^2*d)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/ 
(a^3*c^3*x^5)
 

3.3.75.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=\text {Timed out} \]

input

integrate((e*(b*x**2+a)/(d*x**2+c))**(1/2)/x**6,x)
 

output

Timed out
 

3.3.75.7 Maxima [F]

\[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=\int { \frac {\sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}{x^{6}} \,d x } \]

input

integrate((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^6,x, algorithm="maxima")
 

output

integrate(sqrt((b*x^2 + a)*e/(d*x^2 + c))/x^6, x)
 

3.3.75.8 Giac [F]

\[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=\int { \frac {\sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}{x^{6}} \,d x } \]

input

integrate((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^6,x, algorithm="giac")
 

output

integrate(sqrt((b*x^2 + a)*e/(d*x^2 + c))/x^6, x)
 

3.3.75.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=\int \frac {\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}}{x^6} \,d x \]

input

int(((e*(a + b*x^2))/(c + d*x^2))^(1/2)/x^6,x)
 

output

int(((e*(a + b*x^2))/(c + d*x^2))^(1/2)/x^6, x)